Unable to move a mirrored part of an object
A mesh data is fed through the modifier stack from top to bottom, and what's accessible through Edit mode is only the original data, not the result of the modifier stack. Modification of a modifier's output is only possible by applying it first, or through another modifier lower down the stack.

In your example, vertices generated on-the-fly by Mirror modifier can't be moved past the modifier's mirror axis, even if Clipping is disabled and the modifier is applied to editing cage. That's because it's not in the original mesh data, and its position is not stored independently but calculated by the modifier:



Modifying those vertices manually is possible after applying it, because those new vertices are now available in the original mesh data. Other modifier can also modify them if placed below the Mirror modifier, for example an Armature modifier (note the modifier stack):



or a Lattice modifier: